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===6.31. Relations in General===

===6.31. Relations in General===

In a realistic computational framework, where incomplete and inconsistent information is the rule, it is necessary to work with genera of relations that are increasingly relaxed in their constraining characters but still preserve a measure of analogy with the fundamental species of relations that are found to be prevalent in perfect information contexts.

In a realistic computational framework, where incomplete and inconsistent information is the rule, it is necessary to work with genera of relations that are increasingly relaxed in their constraining characters but still preserve a measure of analogy with the fundamental species of relations that are found to be prevalent in perfect information contexts.

In the present application the kinds of relations of primary interest are functions, equivalence relations, and other species of relations defined by axiomatic properties.  Thus, the information theoretic generalizations of these structures lead to partially defined functions and partially constrained versions of these specially defined classes of relations.

In the present application the kinds of relations of primary interest are functions, equivalence relations, and other species of relations defined by axiomatic properties.  Thus, the information-theoretic generalizations of these structures lead to partially defined functions and partially constrained versions of these specially defined classes of relations.

The purpose of this Section is to outline the kinds of generalized functions and other families of relations that are needed to extend the discussion of the present example.  In this connection, to frame the problem in concrete terms, I need to adapt the square bracket notation for two generalizations of equivalence relations, to be defined below.  But first, a number of broader issues need to be treated.

The purpose of this Section is to outline the kinds of generalized functions and other families of relations that are needed to extend the discussion of the present example.  In this connection, to frame the problem in concrete terms, I need to adapt the square bracket notation for two generalizations of equivalence relations, to be defined below.  But first, a number of broader issues need to be treated.

Generally speaking, one is free to interpret references to generalized objects either as indications of partially formed analogues or as partially informed descriptions of their corresponding species.  I refer to these alternatives as the "object theoretic" and the "sign theoretic" options, respectively.  The first interpretation assumes that vague and general references still have denotations, merely to vague and general objects.  The second interpretation ascribes the partialities of information to the characters of the signs and expressions that are doing the denoting.  In most cases that arise in casual discussion the choice between these conventions is purely stylistic.  However, in many of the more intricate situations that arise in formal discussion the object choice often fails utterly, and whenever the utmost care is required it will usually be the attention to signs that saves the day.

Generally speaking, one is free to interpret references to generalized objects either as indications of partially formed analogues or as partially informed descriptions of their corresponding species.  I refer to these alternatives as the ''object-theoretic'' and the ''sign-theoretic'' options, respectively.  The first interpretation assumes that vague and general references still have denotations, merely to vague and general objects.  The second interpretation ascribes the partialities of information to the characters of the signs and expressions that are doing the denoting.  In most cases that arise in casual discussion the choice between these conventions is purely stylistic.  However, in many of the more intricate situations that arise in formal discussion the object choice often fails utterly, and whenever the utmost care is required it will usually be the attention to signs that saves the day.

In order to speak of generalized orders of relations I need to outline the dimensions of variation along which I intend the characters of already familiar orders of relations to be broadened.  Generally speaking, the taxonomic features of n place relations that I wish to liberalize can be read off from their "local incidence properties" (LIPs).

In order to speak of generalized orders of relations I need to outline the dimensions of variation along which I intend the characters of already familiar orders of relations to be broadened.  Generally speaking, the taxonomic features of <math>n\!</math>-place relations that I wish to liberalize can be read off from their ''local incidence properties'' (LIPs).

Definition.  A "local incidence property" of an n place relation R is one that is based on the following sorts of data.  Suppose R c X1x...xXn.  Pick an element x in one of the domains Xi of R.  Let "R&x@i" denote a subset of R called the "flag of R with x at i", or the "x@i flag of R".  The "local flag" R&x@i c R is defined as follows:

Definition.  A "local incidence property" of an n place relation R is one that is based on the following sorts of data.  Suppose R c X1x...xXn.  Pick an element x in one of the domains Xi of R.  Let "R&x@i" denote a subset of R called the "flag of R with x at i", or the "x@i flag of R".  The "local flag" R&x@i c R is defined as follows: